Page 103 - 35Linear Algebra

5.1 Examples of Vector Spaces 103

Example of a vector space

Example 58
N
R = {f | f : N → R}
Here the vector space is the set of functions that take in a natural number n and return
a real number. The addition is just addition of functions: (f 1 +f 2 )(n) = f 1 (n)+f 2 (n).
Scalar multiplication is just as simple: c · f(n) = cf(n).
We can think of these functions as infinitely large ordered lists of numbers: f(1) =
3
3
1 = 1 is the first component, f(2) = 2 = 8 is the second, and so on. Then for
3
example the function f(n) = n would look like this:
 
1
 8 
 
27
 
 
f =  .  .
.
 . 
 
 n 3 
 
. . .
N
Thinking this way, R is the space of all infinite sequences. Because we can not write
a list infinitely long (without infinite time and ink), one can not define an element of
this space explicitly; definitions that are implicit, as above, or algebraic as in f(n) = n 3
(for all n ∈ N) suffice.
Let’s check some axioms.

(+i) (Additive Closure) (f 1 + f 2 )(n) = f 1 (n) + f 2 (n) is indeed a function N → R,
since the sum of two real numbers is a real number.

(+iv) (Zero) We need to propose a zero vector. The constant zero function g(n) = 0
works because then f(n) + g(n) = f(n) + 0 = f(n).

The other axioms should also be checked. This can be done using properties of the
real numbers.

Reading homework: problem 1

Example 59 The space of functions of one real variable.

R = {f | f : R → R}
R

103